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9.6 π Approximation

approximation We can approximate the number π using the formula:

┌│ ---∘----------------------- │∘ ∘ ------∘-----√--- π ≈ 2k 2- 2 + 2 + ... 2 + 2
with k the number of squareroots. The greater is k, the better is the π .

The formula contains the recursive expression 2 + ∘ -------∘----√--- 2 + ... 2+ 2, so let’s code:
# k is the number of squareroots  
to approxpi :k  
write "approximation:\  print (power 2 :k) * squareroot (2- squareroot (calc :k-2))  
print "-------------------------  
write "pi:\  print pi  
end  
 
to calc :p  
if :p=0 [output 2][output 2+squareroot calc :p-1]  
end  
 
approxpi 10  
Approximation: 3.141591421568446  
-------------------------  
Pi: 3.141592653589793

We found the first 5 digits! If we’re looking for more π digits, we have to allow a better precision with a higher number of digits while computing. Thus, we’re going to use the primitive setdigits.

setdigits 100  
approxpi 100  
Approximation: 3.1415926535897932384626433832795028841973393069670160975807684313880468...  
-------------------------  
Pi: 3.141592653589793238462643383279502884197169399375105820974944592307816406....

And now, we have 39 digits...

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